Add warm water to a small bowl or measuring cup. Stir in yeast and 1 teaspoon sugar. Let rest until foamy, about 5-10 minutes.
In the bowl of a stand mixer fitted with the dough hook, add 4 1/2 cups of flour, remaining sugar, and salt. Turn the mixer on low and mix for 10 seconds to combine.
In a separate bowl, whisk together melted butter, milk, vanilla extract, and 2 eggs.
With the mixer on low (speed 2) add the yeast mixture to the bowl of the mixer. Slowly pour in the butter mixture. Continue to mix on low, scraping the flour from the sides of the bowl as you go. Once all of the flour is incorporated into the dough, slowly add an additional 1/2 cup of flour, scraping the sides of the bowl as needed.
Stop the mixer and remove dough from the hook and scrape down the sides and bottom of the bowl. Turn the mixer on low (speed 2) and gradually add the chocolate chips. Continue to mix on low for 5 minutes.
Transfer the dough to a lightly floured surface. Continue to knead the dough by hand, gradually adding the remaining 1/4 cup of flour to the dough. The process will take an additional 5 minutes. You'll know the dough is ready when it's elastic and no longer sticky.
Place the dough in a large lightly greased bowl. Cover the bowl with a clean, dry towel and let rise for 1 1/2 hours.
Grease a 9x13-inch pan with butter or nonstick baking spray; set aside.
Gently punch down the dough. Transfer the dough to a lightly floured surface and divide into 15 equal pieces. Shape each piece into a smooth ball.
Place the dough balls in prepared pan, spacing them 2 inches apart. You should end up with 5 rows, each row made up of 3 dough balls. Cover the pan with a clean, dry towel; let rise for 1 to 1 1/2 hours, or until double in size.
Preheat the oven to 350°F.
Make an egg wash by whisking the remaining egg with 1 teaspoon of water in a small bowl. Gently brush the tops of the buns with the egg wash. Bake the buns for 28-30 minutes, or until the buns are a deep golden brown.
Place the buns on a wire rack and allow them to cool completely in the pan. Pipe the tops of the buns with the icing (see below) to resemble a cross.
Make icing: Whisk together powdered sugar and cinnamon in a large bowl. Whisk in 2 tablespoons of milk. You want the mixture to be thick enough to pipe. If the mixture is too thick, gradually whisk in the extra tablespoon of milk until smooth. Transfer the mixture to a piping bag or large zip-close bag. Snip off the end of the bag.